Math Puzzle

No. In any group of friends, the number of people with an odd number of friends always comes out even. There are 1729 people (odd); people who don't have a hat (odd friends) = necessarily an even number; so, odd (1729) - even = odd; the total number of blue hats (even friends) must necessarily be odd. Keep in mind that Alex saw 163 blue hats: 163 is an odd number, so the limit has already been reached. If he added one more hat, the total would be 164, which is even, which is unsuitable

No. Forgive me, I'm not usually a puzzle solver and I am likely wrong but here I go. Given that each friendship is mutual, we would expect that the total number of friendships (imagine them as arrows between each other) is an even number, because it must be a multiple of 2. Then it follows that if Alex was wearing a blue hat, there would be an odd total number of relations (arrows between each friend such that one friendship = 2 relations) because there would be an odd number of individuals with an odd number of friendships.